Products and Coproducts

We start with the notion of a coproduct. Although it is dual to product on the theoretical level, it is easier to construct than the product in most concrete cases in algebra, topology, and combinatorics. 

Intuitively, one should think of the coproduct of a and b as the way to combine a and b without any additional constraints. In many situations this 

Reversing all arrows, we obtain the dual notion of a product. 

This time around, the intuition for the product of a and b should be that of a direct product. The structure morphisms should be thought of as corresponding projection maps. 

The special case of products and coproducts in a partially ordered set regarded as a category leads to an important notion. 

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The term feedstock in this article refers not only to coal, but also to products and coproducts of coal conversion processes used to meet the raw material needs of the chemical industry. This definition distinguishes between use of coal-derived products for fuels and for chemicals, but this distinction is somewhat arbitrary because the products involved in fuel and chemical appHcations are often identical or related by simple transformations. For example, methanol has been widely promoted and used as a component of motor fuel, but it is also used heavily in the chemical industry. Frequendy, some or all of the chemical products of a coal conversion process are not isolated but used as process fuel. This practice is common in the many coke plants that are now burning coal tar and naphtha in the ovens. 

Since dehydrogenase catalysis is reversible, the production enzyme can be used to perform the regeneration reaction of NAD(P)+ using a suitable cosubstrate as electron acceptor. In this case, the regeneration enzyme in Fig. 16.2-2 is identical with the production dehydrogenase. However, conversion rates in this set-up tend to be low because of a given reaction equilibrium, which requires an efficient method to withdraw the products and coproducts. 

Another problem is the detoxification of the hazardous wastes that are already present in the environment. Efforts are thwarted by the problan of how less toxic are the detoxification products in themselves Thus, in incineration, for instance, what are the toxicides of all the final combustion products Called products of incomplete combustion, or PlCs, these are the myriad by-products and coproducts of the competing reactions that occur during combustion, and for that matter, during any chemical conversion, more or less. Can they ever be fully detected and analyzed Or if selectively scrubbed, what is to be the disposition of the absorbed materials What is called detoxification may be merely a further dispersion throughout the ecosphere, and a process of trading one set of problems for another. 

The environmental, economic, and social impacts are allocated for main products and coproducts (e.g., rice and rice husk in the rice husk preparation process, electricity and ash in the electricity generation process) according to their economic values. It should be noted that in this case study the main product (electricity) is quantified in MWh while its coproducts (bottom ash, fly ash, etc.) are quantified in tonne. The differences in quantification units make it impossible to allocate environmental, social, and economic impacts by weight basis. Among the available allocation methods, though the economic-based allocation is Hmited because of changing market values, it is, in this case study, the most suitable choice for assessing the sustainability of the product system. 

Definition 4.22. A poset P is called a lattice if when regarded as a category, it has products and coproducts. 

Initial and terminal objects, as well as products and coproducts, are special cases of more general constructions of limits and colimits. The latter are defined and investigated in Section 4.4. 

The natural generalization of the notion of a lattice would be to require that the acyclic category have all finite products and coproducts. Unfortunately, this does not bring an3d,hing new, as the next proposition shows. 

This book covers recent developments in process engineering and resource conservation for biomass conversion systans at scales ranging from the molecular level aU the way to macrolevel supply chains. It provides an overview of process development in biomass conversion systems, with focus on biorefineries involving the production and coproduction of fuels, heating, cooling, and chemicals. Various techniques for enhancing the efficiency... 

The tail and thru stock streams are then immediately dehydrated to decrease the moisture to approximately 14%. At this moisture, the refined grits and their products and coproducts will have prolonged shelf life. The drying operation is usually... 

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